Quasi-Monte Carlo Methods: What, Why, and How?

TL;DR

Quasi-Monte Carlo methods replace IID sampling with low-discrepancy sequences to accelerate convergence in estimating $μ = \mathbb{E}(Y)$. The paper surveys LD sequence constructions (lattice, digital nets, Halton), randomization techniques, discrepancy measures via RKHS, stopping criteria, and problem reformulation strategies, highlighting how to maximize performance gains. It emphasizes reducing effective dimension through variable transforms, control variates, and multilevel methods, enabling near $O(1/ε)$ sampling costs under favorable conditions. The work also discusses extensions beyond mean estimation to density estimation, PDEs with uncertainty, and applications in graphics, robotics, and ML, and points to practical software implementations for practitioners.

Abstract

Many questions in quantitative finance, uncertainty quantification, and other disciplines are answered by computing the population mean, $μ:= \mathbb{E}(Y)$, where instances of $Y:=f(\boldsymbol{X})$ may be generated by numerical simulation and $\boldsymbol{X}$ has a simple probability distribution. The population mean can be approximated by the sample mean, $\hatμ_n := n^{-1} \sum_{i=0}^{n-1} f(\boldsymbol{x}_i)$ for a well chosen sequence of nodes, $\{\boldsymbol{x}_0, \boldsymbol{x}_1, \ldots\}$ and a sufficiently large sample size, $n$. Computing $μ$ is equivalent to computing a $d$-dimensional integral, $\int f(\boldsymbol{x}) \varrho(\boldsymbol{x}) \, \mathrm{d} \boldsymbol{x}$, where $\varrho$ is the probability density for $\boldsymbol{X}$. Quasi-Monte Carlo methods replace independent and identically distributed sequences of random vector nodes, $\{\boldsymbol{x}_i \}_{i = 0}^{\infty}$, by low discrepancy sequences. This accelerates the convergence of $\hatμ_n$ to $μ$ as $n \to \infty$. This tutorial describes low discrepancy sequences and their quality measures. We demonstrate the performance gains possible with quasi-Monte Carlo methods. Moreover, we describe how to formulate problems to realize the greatest performance gains using quasi-Monte Carlo. We also briefly describe the use of quasi-Monte Carlo methods for problems beyond computing the mean, $μ$.

Figures (11)

• Figure 1: The relative error of approximating $\mu$ defined in \ref{['eq:keisterC']} by the sample mean, $\hat{\mu}_n$, defined in \ref{['eq:sample_mean_Keister']} for various choices of nodes, ${\boldsymbol{x}}_0, {\boldsymbol{x}}_1, \ldots$. Grids have the largest error and LD nodes have the smallest error.
• Figure 4: LD nodes cover the unit cube even better than IID nodes or grids. In any projection there is a similar looking distribution of all $64$ nodes.
• Figure 5: The construction of a good lattice node set in two dimensions with $16$ nodes is obtained by moving ${\boldsymbol{h}}/n$ beyond the present node and wrapping around the boundaries of the square (hypercube in $d$ dimensions) until one returns to the origin.
• Figure 6: An extensible lattice corresponding to Figure \ref{['fig:latticeconstruct']} with the nodes reordered using the van der Corput sequence in base $2$. For each plot the blue dots are a copy of the nodes to the left and the orange diamonds are a shifted copy (modulo ${\boldsymbol{1}}$) of the blue dots.
• Figure 7: Each box in the two dimensional projections of the node set \ref{['eq:smallnet']} plotted above contain one node except for the second row, third plot from the left. Thus, this node set is not a $(0,3,3)$-net. It is however a $(1,3,3)$-net.